Rainbow arborescence in random digraphs
Deepak Bal, Patrick Bennett, Colin Cooper, Alan Frieze, Pawe{\l}, Pra{\l}at

TL;DR
This paper proves that in a random directed graph process with randomly coloured edges, a rainbow arborescence almost surely exists under certain conditions, combining probabilistic analysis with graph theory.
Contribution
It establishes the asymptotic almost sure existence of a rainbow arborescence in a randomly coloured Erdős-Rényi directed graph process.
Findings
Rainbow arborescence exists asymptotically almost surely.
Conditions on the number of colours and edges are sufficient.
The process combines random graph evolution with edge colouring constraints.
Abstract
We consider the Erd\H{o}s-R\'enyi random directed graph process, which is a stochastic process that starts with vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let be a graph with edges obtained after steps of this process. Each edge () of independently chooses a colour, taken uniformly at random from a given set of colours. We stop the process prematurely at time when the following two events hold: has at most one vertex that has in-degree zero and there are at least distinct colours introduced ( if at the time when all edges are present there are still less than colours introduced; however, this does not happen asymptotically almost surely). The…
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